If function f and g given by $$f(x)$$ $$=log$$⁡(x$$−$$1)−log⁡(x$$−$$2) and $$g(x)=log$$⁡x−$$1x$$−$$2$$ are equal then x lies in the interval

Question

If function  f   and  g given by $$f(x)$$ $$=log$$⁡(x$$−$$1)−log⁡(x$$−$$2) and $$g(x)=log$$⁡x−$$1x$$−$$2$$  are equal then x lies in the interval

Infinity Learn · Accepted Answer

$$f(x)$$  is defined for all x satisfying x−$$1$$>$$0$$  and  x−$$2$$>$$0$$ i.e. x>$$2$$ Domain (f)=(2$$,∞$$)                       .......              (i)gx$$  is defined for all x satisfyingx−$$1x$$−$$2$$>$$0$$⇒x∈$$($$−∞,$$1)∪(2$$,∞$$)Domain$$ $$(g)=($$−∞,$$1)∪(2$$,∞$$)      ……….(ii) Thus $$f(x)$$  and  $$g(x)$$ are equal for all x belonging to theircommon domain i.e  (2$$,∞$$)

If function $f$ and $g$ given by $f (x) = \log (x - 1) - \log (x - 2)$ and $g (x) = \log (\frac{x - 1}{x - 2})$ are equal then x lies in the interval

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If function f and g given by f(x) =log⁡(x−1)−log⁡(x−2) and g(x)=log⁡x−1x−2 are equal then x lies in the interval

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If function $f$ and $g$ given by $f (x) = \log (x - 1) - \log (x - 2)$ and $g (x) = \log (\frac{x - 1}{x - 2})$ are equal then x lies in the interval